Modular curve 72.432.10-36.bc.1.3

• Label: 72.432.10-36.bc.1.3
• Level: $72$
• Index: $432$
• Genus: $10$
• Cusps: $18$
• $\Q$-cusps: $0$

Invariants

Level:	$72$		$\SL_2$-level:	$36$		Newform level:	$1296$
Index:	$432$		$\PSL_2$-index:	$216$
Genus:	$10 = 1 + \frac{ 216 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 18 }{2}$
Cusps:	$18$ (none of which are rational)		Cusp widths	$6^{9}\cdot18^{9}$		Cusp orbits	$3^{2}\cdot6^{2}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	not computed
$\Q$-gonality:	$4 \le \gamma \le 18$
$\overline{\Q}$-gonality:	$4 \le \gamma \le 10$
Rational cusps:	$0$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:

18G10

Level structure

$\GL_2(\Z/72\Z)$-generators:	$\begin{bmatrix}2&31\\9&16\end{bmatrix}$, $\begin{bmatrix}12&7\\59&4\end{bmatrix}$, $\begin{bmatrix}38&55\\11&60\end{bmatrix}$, $\begin{bmatrix}48&53\\61&44\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 36.216.10.bc.1 for the level structure with $-I$)
Cyclic 72-isogeny field degree:	$12$
Cyclic 72-torsion field degree:	$288$
Full 72-torsion field degree:	$13824$

Models

Canonical model in $\mathbb{P}^{ 9 }$ defined by 28 equations

$ 0 $	$=$	$ y u - w r - t v - t r $
	$=$	$x^{2} + x y - x v + z^{2} - z a - s a$
	$=$	$x^{2} + x y - x r + z^{2} - z s + z a + s a - a^{2}$
	$=$	$x^{2} + x y + x v + x r + z^{2} + z s - s^{2} + s a$
	$=$	$\cdots$

Singular plane model Singular plane model

$ 0 $

$=$

$ x^{9} z^{9} - 27 x^{7} y^{8} z^{3} - 27 x^{7} y^{6} z^{5} - 9 x^{7} y^{4} z^{7} - 18 x^{7} y^{2} z^{9} + \cdots + 27 y^{6} z^{12} $

Rational points

This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.

Maps to other modular curves

Map of degree 3 from the canonical model of this modular curve to the canonical model of the modular curve 36.72.4.m.1 :

$\displaystyle X$	$=$	$\displaystyle -x$
$\displaystyle Y$	$=$	$\displaystyle -u$
$\displaystyle Z$	$=$	$\displaystyle -y$
$\displaystyle W$	$=$	$\displaystyle -z$

Equation of the image curve:

$0$	$=$	$ 3XZ-YW $
	$=$	$ 3X^{3}-XY^{2}+3Z^{3}-9ZW^{2} $

Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 36.216.10.bc.1 :

$\displaystyle X$	$=$	$\displaystyle a$
$\displaystyle Y$	$=$	$\displaystyle y$
$\displaystyle Z$	$=$	$\displaystyle \frac{1}{3}u$

Equation of the image curve:

$0$

$=$

$ 27X^{6}Y^{12}-54X^{4}Y^{14}+27X^{2}Y^{16}-Y^{18}+18XY^{16}Z-54X^{4}Y^{12}Z^{2}-27X^{2}Y^{14}Z^{2}-27X^{7}Y^{8}Z^{3}+54X^{5}Y^{10}Z^{3}+3X^{3}Y^{12}Z^{3}-45XY^{14}Z^{3}+72X^{4}Y^{10}Z^{4}+81X^{2}Y^{12}Z^{4}-27X^{7}Y^{6}Z^{5}+108X^{5}Y^{8}Z^{5}-45XY^{12}Z^{5}+51X^{6}Y^{6}Z^{6}-180X^{4}Y^{8}Z^{6}+135X^{2}Y^{10}Z^{6}+18Y^{12}Z^{6}-9X^{7}Y^{4}Z^{7}-81X^{3}Y^{8}Z^{7}-27XY^{10}Z^{7}-180X^{4}Y^{6}Z^{8}+27X^{2}Y^{8}Z^{8}+X^{9}Z^{9}-18X^{7}Y^{2}Z^{9}+27X^{5}Y^{4}Z^{9}-63X^{3}Y^{6}Z^{9}-81XY^{8}Z^{9}+108X^{4}Y^{4}Z^{10}+243X^{2}Y^{6}Z^{10}-18X^{7}Z^{11}+135X^{5}Y^{2}Z^{11}-81X^{3}Y^{4}Z^{11}-81XY^{6}Z^{11}+18X^{6}Z^{12}-162X^{4}Y^{2}Z^{12}+162X^{2}Y^{4}Z^{12}+27Y^{6}Z^{12}+81X^{5}Z^{13}-81XY^{4}Z^{13}-162X^{4}Z^{14}+81X^{3}Z^{15} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
72.144.4-36.m.1.2	$72$	$3$	$3$	$4$	$?$
72.216.4-18.h.1.1	$72$	$2$	$2$	$4$	$?$
72.216.4-18.h.1.6	$72$	$2$	$2$	$4$	$?$